{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:50:13Z","timestamp":1777449013837,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"3-4","license":[{"start":{"date-parts":[[2012,12,1]],"date-time":"2012-12-01T00:00:00Z","timestamp":1354320000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2012,12]]},"abstract":"<jats:p>\n                    Let u\n                    <jats:sub>\u03b5<\/jats:sub>\n                    be the solution of the Poisson equation in a domain periodically perforated along a manifold \u03b3=\u03a9\u2229{x\n                    <jats:sub>1<\/jats:sub>\n                    =0}, with a nonlinear Robin type boundary condition on the perforations (the flux here being O(\u03b5\n                    <jats:sup>\u2212\u03ba<\/jats:sup>\n                    )\u03c3(x,u\n                    <jats:sub>\u03b5<\/jats:sub>\n                    )), and with a Dirichlet condition on \u2202\u03a9. \u03a9 is a domain of R\n                    <jats:sup>n<\/jats:sup>\n                    with n\u22653, the small parameter \u03b5, that we shall make to go to zero, denotes the period, and the size of each cavity is O(\u03b5\n                    <jats:sup>\u03b1<\/jats:sup>\n                    ) with \u03b1\u22651. The function \u03c3 involving the nonlinear process is a C\n                    <jats:sup>1<\/jats:sup>\n                    (\u03a9\n                    <jats:sup>\u00af<\/jats:sup>\n                    \u00d7R) function and the parameter \u03ba\u2208R. Depending on the values of \u03b1 and \u03ba, the effective equations on \u03b3 are obtained; we provide a critical relation between both parameters which implies a different average of the process on \u03b3 ranging from linear to nonlinear. For each fixed \u03ba a critical size of the cavities which depends on n is found. As \u03b5\u21920, we show the convergence of u\n                    <jats:sub>\u03b5<\/jats:sub>\n                    in the weak topology of H\n                    <jats:sup>1<\/jats:sup>\n                    and construct correctors which provide estimates for convergence rates of solutions. All this allows us to derive convergence for the eigenelements of the associated spectral problems in the case of \u03c3 a linear function.\n                  <\/jats:p>","DOI":"10.3233\/asy-2012-1116","type":"journal-article","created":{"date-parts":[[2019,11,29]],"date-time":"2019-11-29T19:24:17Z","timestamp":1575055457000},"page":"289-322","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":20,"title":["On homogenization of nonlinear Robin type boundary conditions for cavities along manifolds and associated spectral problems"],"prefix":"10.1177","volume":"80","author":[{"given":"D.","family":"G\u00f3mez","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Estad\u00edstica y Computaci\u00f3n, Universidad de Cantabria, Santander, Spain"}]},{"given":"E.","family":"P\u00e9rez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada y Ciencias de la Computaci\u00f3n, Universidad de Cantabria, Santander, Spain"}]},{"given":"T.A.","family":"Shaposhnikova","sequence":"additional","affiliation":[{"name":"Department of Differential Equations, Moscow State University, Moscow, Russia"}]}],"member":"179","published-online":{"date-parts":[[2012,12,1]]},"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2012-1116","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2012-1116","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:39:45Z","timestamp":1777379985000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-2012-1116"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,12]]},"references-count":0,"journal-issue":{"issue":"3-4","published-print":{"date-parts":[[2012,12]]}},"alternative-id":["10.3233\/ASY-2012-1116"],"URL":"https:\/\/doi.org\/10.3233\/asy-2012-1116","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,12]]}}}